Iterative proportional fitting

The iterative proportional fitting procedure (IPF or IPFP, also known as biproportional fitting or biproportion in statistics or economics (input-output analysis, etc.), RAS algorithm^[1] in economics, raking in survey statistics, and matrix scaling in computer science) is the operation of finding the fitted matrix ${\displaystyle X}$ which is the closest to an initial matrix ${\displaystyle Z}$ but with the row and column totals of a target matrix ${\displaystyle Y}$ (which provides the constraints of the problem; the interior of ${\displaystyle Y}$ is unknown). The fitted matrix being of the form ${\displaystyle X=PZQ}$, where ${\displaystyle P}$ and ${\displaystyle Q}$ are diagonal matrices such that ${\displaystyle X}$ has the margins (row and column sums) of ${\displaystyle Y}$. Some algorithms can be chosen to perform biproportion. We have also the entropy maximization,^[2][3] information loss minimization (or cross-entropy)^[4] or RAS which consists of factoring the matrix rows to match the specified row totals, then factoring its columns to match the specified column totals; each step usually disturbs the previous step's match, so these steps are repeated in cycles, re-adjusting the rows and columns in turn, until all specified marginal totals are satisfactorily approximated. However, all algorithms give the same solution.^[5] In three- or more-dimensional cases, adjustment steps are applied for the marginals of each dimension in turn, the steps likewise repeated in cycles.